A prime means a derivative with respect to \(N=\ln a\) , with \(a\)  the scale factor. This system is studied in [1]} where it is shown that all its equilibrium points are such as \(\Omega _m\)  or/and \(\Omega _r=0\) . Hence General Relativity with non coupled matter, radiation and dark energy cannot produce a homoclinic orbit that needs a center equilibrium point with \(\Omega _i\ne 0\) . It is thus not compatible with a \(\Lambda CDM\)  periodic cosmology. This is also true for any extended theories of gravity that can be rewritten as General Relativity with an effective dark energy[2]}. For instance, let us consider an \(f(R)\)  theory in the Palatini formalism. Using [3]}, we define the following normalised variables
\(\Omega _m=\frac{k}{3}\frac{\rho _m}{3F\xi H^2}\) 
\(\Omega _r=\frac{k}{3}\frac{\rho _r}{3F\xi H^2}\) 
\(\Omega _f=\frac{FR-f}{6F\xi H^2}\) 
